WaveletMatrixSum.go

// !deprecated
//
// 维护区间贡献的 Wavelet Matrix
// !注意查询区间贡献时, 异或无效

// CountRange(start, end, a, b, xor) - 区间 [start, end) 中值在 [a, b) 之间的数的个数和这些数的和.
// CountPrefix(start, end, x, xor) - 区间 [start, end) 中值在 [0, x) 之间的数的个数和这些数的和.

// Kth(start, end, k, xor) - 区间 [start, end) 中第 k 小的数(0-indexed) 和前 k 小的数的和(不包括这个数).

// Floor(start, end, x, xor) - 区间 [start, end) 中值小于等于 x 的最大值
// Ceiling(start, end, x, xor) - 区间 [start, end) 中值大于等于 x 的最小值

// MaxRightValue(start, end, xor, check) - 返回使得 check(prefixSum) 为 true 的最大value, 其中prefixSum为[0,val)内的数的和.
// MaxRightCount(start, end, xor, check) - 返回使得 check(prefixSum) 为 true 的区间前缀个数的最大值.

package main

import (
	"bufio"
	"fmt"
	"math"
	"math/bits"
	"os"
	"sort"
	"time"
)

func main() {

	// 区间前驱后继()
	// demo()
	CF1771F()
	// 区间最短距离和()
}

// https://yukicoder.me/problems/no/1332
func 区间前驱后继() {
	in := bufio.NewReader(os.Stdin)
	out := bufio.NewWriter(os.Stdout)
	defer out.Flush()

	var n int32
	fmt.Fscan(in, &n)
	nums := make([]int, n)
	for i := int32(0); i < n; i++ {
		fmt.Fscan(in, &nums[i])
	}

	wm := NewWaveletMatrixSum(nums, -1, func() int { return 0 }, func(a, b int) int { return a + b }, func(a int) int { return -a }, nil)
	var q int32
	fmt.Fscan(in, &q)
	for i := int32(0); i < q; i++ {
		var start, end, x int
		fmt.Fscan(in, &start, &end, &x)
		start--

		res := math.MaxInt
		floor := wm.Floor(start, end, x, 0)
		if floor != -INF {
			res = min(res, abs(x-floor))
		}
		ceil := wm.Ceiling(start, end, x, 0)
		if ceil != INF {
			res = min(res, abs(x-ceil))
		}
		fmt.Fprintln(out, res)
	}
}

// Hossam and Range Minimum Query
// https://www.luogu.com.cn/problem/CF1771F
// 在线查询区间出现次数为奇数的数的最小值.
//
// 异或哈希+Wavelet Matrix二分: 最大值<x时，区间内的异或和是否为0(出现次数全是偶数次).
func CF1771F() {
	in := bufio.NewReader(os.Stdin)
	out := bufio.NewWriter(os.Stdout)
	defer out.Flush()

	var n int
	fmt.Fscan(in, &n)
	nums := make([]int, n)
	for i := 0; i < n; i++ {
		fmt.Fscan(in, &nums[i])
	}
	getRank, getValue, size := DiscretizeCompressed(nums, 0)
	for i := 0; i < n; i++ {
		nums[i] = getRank(nums[i])
	}
	R := NewRandom()
	xorHash := make([]int, size)
	for i := 0; i < size; i++ {
		xorHash[i] = int(uint32(R.Rng()))
	}
	wm := NewWaveletMatrixSum(
		nums, -1,
		func() E { return 0 },
		func(a, b E) E { return a ^ b },
		func(a E) E { return a },
		func(v int) E { return xorHash[v] },
	)

	preRes := 0
	normalize := func(v int) int {
		return v ^ preRes
	}

	var q int
	fmt.Fscan(in, &q)
	for i := 0; i < q; i++ {
		var left, right int
		fmt.Fscan(in, &left, &right)
		left = normalize(left)
		right = normalize(right)
		left--
		res := wm.MaxRightValue(left, right, 0, func(preSum E) bool { return preSum == 0 })
		if res == INF {
			res = 0
		} else {
			res = getValue(res)
		}
		preRes = res
		fmt.Fprintln(out, res)
	}
}

func 区间最短距离和() {
	// https://yukicoder.me/problems/no/924
	// n,q<=2e5
	// -1e9 <= nums[i] <= 1e9
	// 给定n个查询[l,r]
	// !求区间[l,r]中位数到区间[l,r]中每个数的距离之和
	// !也就求函数 f(x)= ∑|nums[i]-x| (l<=i<=right) 的最小值
	// !区间中位数
	in := bufio.NewReader(os.Stdin)
	out := bufio.NewWriter(os.Stdout)
	defer out.Flush()

	var n, q int
	fmt.Fscan(in, &n, &q)
	OFFSET := int(1e9 + 10)
	nums := make([]int, n)
	for i := range nums {
		fmt.Fscan(in, &nums[i])
		nums[i] += OFFSET
	}
	preSum := make([]int, n+1)
	for i := range nums {
		preSum[i+1] = preSum[i] + nums[i]
	}

	wm := NewWaveletMatrixSum(nums, -1, func() int { return 0 }, func(a, b int) int { return a + b }, func(a int) int { return -a }, nil)
	for i := 0; i < q; i++ {
		var left, right int
		fmt.Fscan(in, &left, &right)
		left--

		n := right - left
		lowerCount := n / 2
		ceilCount := n - lowerCount
		mid, lowerSum := wm.Kth(left, right, lowerCount, 0)
		_, allSum := wm.Kth(left, right, n, 0)
		ceilSum := allSum - lowerSum

		res := 0
		res += mid*lowerCount - lowerSum
		res += ceilSum - mid*ceilCount
		fmt.Fprintln(out, res)
	}
}

func abc281_e() {
	in := bufio.NewReader(os.Stdin)
	out := bufio.NewWriter(os.Stdout)
	defer out.Flush()
	var n, m, k int
	fmt.Fscan(in, &n, &m, &k)
	nums := make([]int, n)
	for i := range nums {
		fmt.Fscan(in, &nums[i])
	}
	wm := NewWaveletMatrixSum(nums, -1, func() int { return 0 }, func(a, b int) int { return a + b }, func(a int) int { return -a }, nil)
	for i := 0; i < n-m+1; i++ {
		_, res := wm.Kth(i, i+m, k, 0)
		fmt.Fprintln(out, res)
	}
}

func demo() {
	nums := []int{3, 1, 2, 4, 5, 6, 7, 8, 9, 10}
	wm := NewWaveletMatrixSum(nums, -1, func() int { return 0 }, func(a, b int) int { return a + b }, func(a int) int { return -a }, nil)
	fmt.Println(wm.CountRange(0, 10, 3, 7, 0))
	fmt.Println(wm.Kth(0, 10, 3, 0))                                                    // 3
	fmt.Println(wm.MaxRightValue(0, 10, 0, func(preSum E) bool { return preSum < 11 })) // 5 即值域在 [0,5) 中的数的和小于 11
	fmt.Println(wm.MaxRightCount(0, 10, 0, func(preSum E) bool { return preSum < 11 })) // 4 即排序后前 4 个数的和小于 11
	fmt.Println(wm.Ceiling(0, 10, 3, 0))
	fmt.Println(wm.Floor(0, 10, 3, 0))
}

const INF int = 1e18

type E = int

type WaveletMatrixSum struct {
	n, log int
	mid    []int
	bv     []*BitVector
	preSum [][]E
	unit   E
	e      func() E
	op     func(a, b E) E
	inv    func(a E) E
}

// log:如果要支持异或,则需要按照异或的值来决定值域
//
//	设为-1时表示不使用异或
func NewWaveletMatrixSum(
	nums []E, log int,
	e func() E, op func(a, b E) E, inv func(a E) E,
	// f: 求和的变换函数.nil表示不变换，即v.
	f func(v E) E,
) *WaveletMatrixSum {
	numsCopy := make([]E, len(nums))
	max_ := 1
	for i, v := range nums {
		numsCopy[i] = v
		if v > max_ {
			max_ = v
		}
	}
	if log == -1 {
		log = bits.Len(uint(max_))
	}
	res := &WaveletMatrixSum{e: e, op: op, inv: inv}
	res.unit = res.e()
	n := len(numsCopy)
	mid := make([]int, log)
	bv := make([]*BitVector, log)
	for i := 0; i < log; i++ {
		bv[i] = NewBitVector(n)
	}
	preSum := make([][]E, log+1)
	for i := range preSum {
		preSum[i] = make([]E, n+1)
		for j := range preSum[i] {
			preSum[i][j] = res.unit
		}
	}

	a0, a1 := make([]E, n), make([]E, n)
	for d := log - 1; d >= -1; d-- {
		p0, p1 := 0, 0
		for i := 0; i < n; i++ {
			tmp := numsCopy[i]
			if f != nil {
				tmp = f(tmp)
			}
			preSum[d+1][i+1] = res.op(preSum[d+1][i], tmp)
		}
		if d == -1 {
			break
		}
		for i := 0; i < n; i++ {
			f := (numsCopy[i] >> d) & 1
			if f == 0 {
				a0[p0] = numsCopy[i]
				p0++
			} else {
				bv[d].Set(i)
				a1[p1] = numsCopy[i]
				p1++
			}
		}
		mid[d] = p0
		bv[d].Build()
		numsCopy, a0 = a0, numsCopy
		for i := 0; i < p1; i++ {
			numsCopy[p0+i] = a1[i]
		}
	}

	res.n, res.log = n, log
	res.mid, res.bv, res.preSum = mid, bv, preSum
	return res
}

// 返回区间 [left, right) 中 范围在 [a, b) 中的 (元素的个数, op 的结果)
func (wm *WaveletMatrixSum) CountRange(left, right, a, b, xor int) (int, E) {
	c1, s1 := wm.CountPrefix(left, right, a, xor)
	c2, s2 := wm.CountPrefix(left, right, b, xor)
	return c2 - c1, wm.op(wm.inv(s1), s2)
}

// 返回区间 [left, right) 中 范围在 [0, x) 中的 (元素的个数, op 的结果)
func (wm *WaveletMatrixSum) CountPrefix(left, right, x, xor int) (int, E) {
	if x >= 1<<wm.log {
		return right - left, wm.get(wm.log, left, right)
	}
	count := 0
	sum := wm.unit
	for d := wm.log - 1; d >= 0; d-- {
		add := (x >> d) & 1
		f := (xor >> d) & 1
		l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0)
		kf := f*(right-left-r0+l0) + (f^1)*(r0-l0)
		if add == 1 {
			count += kf
			if f == 1 {
				sum = wm.op(sum, wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0))
				left, right = l0, r0
			} else {
				sum = wm.op(sum, wm.get(d, l0, r0))
				left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
			}
		} else if f == 0 {
			left, right = l0, r0
		} else {
			left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
		}
	}

	return count, sum
}

// 返回区间 [left, right) 中的 (第k小的元素, 前k个元素(不包括第k小的元素) 的 op 的结果)
//
//	如果k < 0, 返回 (-1, 0); 如果k >= right-left, 返回 (-1, 区间 op 的结果)
func (wm *WaveletMatrixSum) Kth(left, right, k, xor int) (int, E) {
	if k < 0 {
		return -1, 0
	}
	if right-left <= k {
		return -1, wm.get(wm.log, left, right)
	}
	res, sum := 0, wm.unit
	for d := wm.log - 1; d >= 0; d-- {
		f := (xor >> d) & 1
		l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0)
		kf := f*(right-left-r0+l0) + (f^1)*(r0-l0)
		if k < kf {
			if f == 0 {
				left, right = l0, r0
			} else {
				left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
			}
		} else {
			k -= kf
			res |= 1 << d
			if f == 1 {
				sum = wm.op(sum, wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0))
				left, right = l0, r0
			} else {
				sum = wm.op(sum, wm.get(d, l0, r0))
				left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
			}
		}
	}

	if k != 0 {
		sum = wm.op(sum, wm.get(0, left, left+k))
	}
	return res, sum
}

// 返回使得 check(prefixSum) 为 true 的最大值 val.
//
//	!(即区间内小于 val 的数的聚合值 prefixSum 满足 check函数, 找到这样的最大的 val)
//	如果整个区间都满足, 返回 INF.
//	eg: val = 5 => 即区间内值域在 [0,5) 中的数的聚合值满足 check 函数.
func (wm *WaveletMatrixSum) MaxRightValue(left, right, xor int, check func(preSum E) bool) E {
	if check(wm.get(wm.log, left, right)) {
		return INF
	}
	res := 0
	sum := wm.unit
	for d := wm.log - 1; d >= 0; d-- {
		f := (xor >> d) & 1
		l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0)
		var loSum E
		if f == 0 {
			loSum = wm.get(d, l0, r0)
		} else {
			loSum = wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0)
		}
		if check(wm.op(sum, loSum)) {
			sum = wm.op(sum, loSum)
			res |= 1 << d
			if f == 1 {
				left, right = l0, r0
			} else {
				left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
			}
		} else if f == 0 {
			left, right = l0, r0
		} else {
			left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
		}
	}

	return res
}

// 返回使得 check(prefixSum) 为 true 的区间前缀个数的最大值.
//
//	eg: count = 4 => 即区间内的数排序后, 前4个数的和满足 check 函数.
func (wm *WaveletMatrixSum) MaxRightCount(left, right, xor int, check func(preSum E) bool) int {
	if check(wm.get(wm.log, left, right)) {
		return right - left
	}

	res := 0
	sum := wm.unit
	for d := wm.log - 1; d >= 0; d-- {
		f := (xor >> d) & 1
		l0, r0 := wm.bv[d].Rank(left, 0), wm.bv[d].Rank(right, 0)
		var kf int
		var loSum E
		if f == 0 {
			kf = r0 - l0
			loSum = wm.get(d, l0, r0)
		} else {
			kf = (right - left) - (r0 - l0)
			loSum = wm.get(d, left+wm.mid[d]-l0, right+wm.mid[d]-r0)
		}

		if check(wm.op(sum, loSum)) {
			sum = wm.op(sum, loSum)
			res += kf
			if f == 1 {
				left, right = l0, r0
			} else {
				left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
			}
		} else if f == 0 {
			left, right = l0, r0
		} else {
			left, right = left+wm.mid[d]-l0, right+wm.mid[d]-r0
		}
	}

	res += wm.binarySearch(func(k int) bool {
		return check(wm.op(sum, wm.get(0, left, left+k)))
	}, 0, right-left)

	return res
}

// [left, right) 中小于等于 x 的数中最大的数
//
//	如果不存在则返回-INF
func (w *WaveletMatrixSum) Floor(start, end, value, xor int) int {
	less, _ := w.CountPrefix(start, end, value, xor)
	if less == 0 {
		return -INF
	}
	res, _ := w.Kth(start, end, less-1, xor)
	return res
}

// [left, right) 中大于等于 x 的数中最小的数
//
//	如果不存在则返回INF
func (w *WaveletMatrixSum) Ceiling(start, end, value, xor int) int {
	less, _ := w.CountPrefix(start, end, value, xor)
	if less == end-start {
		return INF
	}
	res, _ := w.Kth(start, end, less, xor)
	return res
}

func (wm *WaveletMatrixSum) binarySearch(f func(E) bool, ok, ng int) int {
	for abs(ok-ng) > 1 {
		x := (ok + ng) >> 1
		if f(x) {
			ok = x
		} else {
			ng = x
		}
	}
	return ok
}

func (wm *WaveletMatrixSum) get(d, l, r int) E {
	return wm.op(wm.inv(wm.preSum[d][l]), wm.preSum[d][r])
}

func abs(a int) int {
	if a < 0 {
		return -a
	}
	return a
}

type BitVector struct {
	bits   []uint64
	preSum []int32
}

func NewBitVector(n int) *BitVector {
	return &BitVector{bits: make([]uint64, n>>6+1), preSum: make([]int32, n>>6+1)}
}

func (bv *BitVector) Set(i int) {
	bv.bits[i>>6] |= 1 << (i & 63)
}

func (bv *BitVector) Build() {
	for i := 0; i < len(bv.bits)-1; i++ {
		bv.preSum[i+1] = bv.preSum[i] + int32(bits.OnesCount64(bv.bits[i]))
	}
}

func (bv *BitVector) Rank(k int, f int) int {
	m, s := bv.bits[k>>6], bv.preSum[k>>6]
	res := int(s) + bits.OnesCount64(m&((1<<(k&63))-1))
	if f == 1 {
		return res
	}
	return k - res
}

// (紧)离散化.
//
//	offset: 离散化的起始值偏移量.
//
//	getRank: 给定一个数，返回它的排名`(offset ~ offset + count)`.
//	count: 离散化(去重)后的元素个数.
func DiscretizeCompressed(nums []int, offset int) (getRank func(value int) int, getValue func(rank int) int, count int) {
	set := make(map[int]struct{}, len(nums))
	for _, v := range nums {
		set[v] = struct{}{}
	}
	count = len(set)
	rank := make([]int, 0, count)
	for v := range set {
		rank = append(rank, v)
	}
	sort.Ints(rank)
	mp := make(map[int]int, count)
	for i, v := range rank {
		mp[v] = i + offset
	}
	getRank = func(v int) int { return mp[v] }
	getValue = func(r int) int { return rank[r-offset] }
	return
}

type Random struct {
	seed     uint64
	hashBase uint64
}

func NewRandom() *Random                 { return &Random{seed: uint64(time.Now().UnixNano()/2 + 1)} }
func NewRandomWithSeed(seed int) *Random { return &Random{seed: uint64(seed)} }

func (r *Random) Rng() uint64 {
	r.seed ^= r.seed << 7
	r.seed ^= r.seed >> 9
	return r.seed
}
func (r *Random) Rng61() uint64 { return r.Rng() & ((1 << 61) - 1) }